Modeling the Real Term Structure

Transcription

1 Modeling the Real Term Structure (Inflation Risk) Chris Telmer May / 23

2 Old school Old school Prices Goods? Real Return Real Interest Rate TIPS Real yields : Model The Fisher equation defines the real interest rate as: i t = r t +E t (π t,t+1 ) where, i t = nominal, one period interest rate r t = real interest rate E t (π t,t+1 ) = expected inflation rate, between t and t+1 Strictly speaking, this equation isn t quite right. There s a cross-product term and an inflation risk term that are omitted (stay tuned). However, don t lose sight of the fact that it does capture some first-order economic intuition. Fisher s idea was that r t is determined by real productivity. The nominal interest rate, i t, is then determined by people s expectations about inflation, which is determined by monetary growth, which is determined, ultimately, by the central bank. End-of-story. This is where we re going. First, we must understand prices. 2 / 23

3 Prices Old school Prices Goods? Real Return Real Interest Rate TIPS Real yields : Model price of goods in units of USD Notation: is USD/Good. This means USD per Good. The / sign does not mean divided by. 1/ is the price of USD in units of goods. It is a Good/USD price. Price of nominal, one-period bond is b 1 t? It is USD/Bond. Remember: the notation x/y means the price of the thing in the denominator in units of the thing in the numerator. Another example: the USD/EUR nominal exchange rate. Price of euros in units of dollars. What is inflation? The rate of change in the price of goods, denoted π t+1 : Either (1+π t+1 ) +1 or π t+1 log(+1 / ) depending on which is more convenient. 3 / 23

4 Goods? Old school Prices Goods? Real Return Real Interest Rate TIPS Real yields : Model What are goods? For simplicity, you can think of this as a single-good world. Food, for instance. In reality, is the price of the CPI basket of consumer goods. Things get substantially more complex with multiple goods. Relative price effects become important. When they do, it becomes difficult to distinguish inflation from changing relative prices. For example, when the price of oil goes up, is this inflation? Not necessarily. Only if it makes go up. If other goods prices go down in response it is natural that they would then may not increase. Inflation is an increase in the overall price of goods in units of money. 4 / 23

5 Real Return Old school Prices Goods? Real Return Real Interest Rate TIPS Real yields : Model What is a real return? Real Return = Consider the real return on a nominal bond: Number of Goods Received Number of Goods Paid A nominal bond costs USD b 1 t. Equivalently, in units of goods, it costs: Goods b1 t A nominal bond pays USD 1.0. Equivalently, in units of goods, it pays: Goods The real return is therefore Real Return = b 1 t+1 5 / 23

6 (continued) Old school Prices Goods? Real Return Real Interest Rate TIPS Real yields : Model Therefore, we can write the real return on the nominal bond as r t+1 : (1+ r t+1 ) = (1+i t )/(1+π t+1 ) (1) Note that, because of inflation, this return is random. 6 / 23

7 Real Interest Rate Old school Prices Goods? Real Return Real Interest Rate TIPS Real yields : Model The Fisher equation: Where does this come from? Equation (1) gives i t = r t +E t (π t,t+1 ) (2) i t = r t+1 +π t+1 + r t+1 π t+1 Suppose that we ignore the cross-product term (we shouldn t unless r and π are small). Take expectations: i t = E t r t+1 +E t π t+1 (3) So, you might think of reconciling the Fisher, (2), with equation (3) by imposing that r t = E t r t+1. But, you d be wrong. Stay tuned. Note that, absent uncertainty, the Fisher equation is the same as equation (3), with the cross-product ignored. But with uncertainty, we can do better... 7 / 23

8 TIPS Old school Prices Goods? Real Return Real Interest Rate TIPS Real yields : Model r t+1 is not what we mean by the real interest rate. What we mean is the return on a bond which pays a riskless real amount. Call this r t. Good news: we can observe r t. TIPS (roughly): Real return? Cost = USD b 1 t Payoff = USD +1 Real Return = +1/ 1/+1 b 1 t/ = 1 b 1 t (1+r t ) TIPS can be thought of as either (i) a risky USD-valued bond costing USD b 1 t and paying USD +1 /, or (ii) a riskless Goods-valued bond costing (Goods b 1 t/ ) and paying (Goods (+1 / )/+1 ) or (Goods 1/ ). 8 / 23

9 Nominal vs Real Yields? Old school Prices Goods? Real Return Real Interest Rate TIPS Real yields : Model So, because of inflation-indexed bonds, we can observe both the nominal and the real yield curves. The short rates are, respectively, i t and r t. How are these things related? The Fisher equation? No!! How do we find out? Asset pricing theory and the pricing kernel? 9 / 23

10 : Model Real Pricing Kernel Nominal valuation Numeraire Change Real kernel Lognormality Risk Aversion Term Structure? : Model 10 / 23

11 Real Pricing Kernel : Model Real Pricing Kernel Nominal valuation Numeraire Change Real kernel Lognormality Risk Aversion Term Structure? Consider a one-period real bond Cost = Goods c 1 t Payoff = Goods 1.0 By no-arbitrage, there exists a pricing kernel, n t+1, which discounts real cash flows (i.e., cash flows denominated in units of goods): q t = E t n t+1 (q t+1 +δ t+1 ), where q and δ are denominated in units of goods. Therefore, for the one-period bond: c 1 t = E t n t / 23

12 Valuing a Nominal Bond : Model Real Pricing Kernel Nominal valuation Numeraire Change Real kernel Lognormality Risk Aversion Term Structure? The real pricing kernel can be used to value a nominal bond. The nominal bond s cash flows can be denominated in real terms: Cost of nominal bond = Goods b1 t Payoff of nominal bond = Goods 1 +1 The real pricing kernel must price this real security: b 1 t 1 = E t n t+1 +1 = b 1 t = E t n t / 23

13 Change of Numeraire : Model Real Pricing Kernel Nominal valuation Numeraire Change Real kernel Lognormality Risk Aversion Term Structure? No arbitrage implies that b 1 t = E t m t+1 and b 1 t = E t n t+1 +1 If I fix the process n t+1 and define m t+1 = n t+1 /+1, then, obviously, both equations will hold. However, if I can find a ε t+1 which is uncorrelated with /+1, and define m t+1 = (n t+1 +ε t+1 ) +1, then both equations will also hold. So, in general, the mapping between m t+1 and n t+1 is not unique. When will it be unique? Complete markets. If markets are complete, then* m t+1 = n t+1 +1 (4) 13 / 23

14 (continued) : Model Real Pricing Kernel Nominal valuation Numeraire Change Real kernel Lognormality Risk Aversion Term Structure? Equation (4) defines a change of numeraire. You will encounter this a lot in quantitative finance. The most common use is for currency pricing, where one important thing which distinguishes domestic and foreign assets is that they are denominated in different numeraires, or currencies. The change of numeraire result is that we move between pricing kernels in different numeraires by multiplying by the rate of change of the relative price of the two numeraires. 14 / 23

15 Pricing Kernel and Real Rate : Model Real Pricing Kernel Nominal valuation Numeraire Change Real kernel Lognormality Risk Aversion Term Structure? No arbitrage: b 1 t = E t n t+1 +1 = E t n t+1 1 (1+π t+1 ) Risk neutrality: n t+1 is a constant. (5) b 1 1 t = ne t (1+π t+1 ) 1 (1+r) E t 1 1+i t = 1 (1+π t+1 ) (6) (7) which is intuitive, but ugly. 15 / 23

16 Lognormality : Model Real Pricing Kernel Nominal valuation Numeraire Change Real kernel Lognormality Risk Aversion Term Structure? Risk neutrality and lognormal inflation Which gives, logn t+1 = r log(+1 / ) = µ π +σ π w t+1, w t+1 N(0,1) b 1 t = E t n t+1 +1 = e r E t +1 = e r µ π+σ 2 π /2 Therefore, i t = r+µ π σ 2 π/2 (8) = r+e t log(+1 / ) σ 2 π/2 This is the Fisher equation with volatility adjustment (an analog of the cross-product from above). Fisher Risk Neutrality 16 / 23

17 Risk Aversion : Model Real Pricing Kernel Nominal valuation Numeraire Change Real kernel Lognormality Risk Aversion Term Structure? logn t+1 = µ n +λε t+1 log(+1 / ) = µ π +σ π w t+1 [ ] ([ ] [ εt β N, 0 β 1 w t+1 Nominal bond price equals conditional mean of nominal kernel or, equivalently conditional mean of numeraire-changed real kernel: b 1 t ]) = E t n t+1 +1 (9) Standard lognormal algebra gives: i = µ n +µ π λ 2 /2 σ 2 π/2 2λσ π β = r +E log(+1 / ) σ 2 π/2 λσ π β This is a modern, arbitrage-free version of Fisher s idea which incorporates an inflation risk premium, σ 2 π/2 λσ π β (I ve dropped the t subscript because, here, everything is i.i.d.). 17 / 23

18 Term Structure? : Model Real Pricing Kernel Nominal valuation Numeraire Change Real kernel Lognormality Risk Aversion Term Structure? The real, one-period holding return on a two-period bond: +1 b 1 t+1 b 2 t Therefore, by the definition of n t+1 b 2 t = E t (n t+1 +1 b 1 t+1) The two-period nominal yield ends up looking like: 2y 2 t = r t +E t r t+1 +E t (π t+1 +π t+1,t+2 )+rp t where π t+1,t+2 log(+2 /+1 ) and rp t is the risk premium. The risk premium depends on term risk and inflation risk and interactions between the two. 18 / 23

19 : Model Appendix: IL Markets IL markets Appendix: Information on IL Markets 19 / 23

20 Inflation-linked bond markets : Model IL markets Some issues faced by IL market participants (WSJ article on BlackBoard) Trading opportunities based on differences between implied and realized inflation. Example: France Government no longer sets interest rate on Livret-A savings accounts. Rate will be indexed to inflation Demand for French IL bonds has taken off, pushing up implied inflation. Intra-European carry trades: sell German IL bonds, buy U.K. IL bonds. Profit will depend on evolution of implied inflation and relation to actual inflation. Key point: Payoffs depends on current and future monetary policy. 20 / 23

21 (continued) : Model IL markets 21 / 23

22 (continued) : Model IL markets 22 / 23

23 (continued) : Model IL markets 23 / 23